Questions C4 (1219 questions)

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OCR MEI C4 Q8
19 marks Standard +0.3
8 Scientists predict the velocity ( \(v\) kilometres per minute) for the new "outer explorer" spacecraft over the first minute of its entry to the atmosphere of the planet Titan to be modelled by the equation: $$v = \frac { 5000 } { ( 1 + t ) ( 2 + t ) ^ { 2 } } , 0 \leq t \leq 1 \text { where } t \text { represents time in minutes. }$$
  1. Use a binomial expansion to expand \(( 1 + t ) ^ { - 1 }\) up to and including the term in \(t ^ { 2 }\).
  2. Use a binomial expansion to expand \(( 2 + t ) ^ { - 2 }\) up to and including the term in \(t ^ { 2 }\).
  3. Hence, or otherwise, show that \(v \approx 1250 \left( 1 - 2 t + \frac { 11 t ^ { 2 } } { 4 } \right)\).
  4. The displacement of the spacecraft can be found by calculating the area under the velocity time graph. Use the approximation found in part (iii) to estimate the displacement of the spacecraft over the first half minute.
  5. Write \(\frac { 1 } { ( 1 + t ) ( 2 + t ) ^ { 2 } }\) in partial fractions.
  6. The displacement of the spacecraft in the first \(T\) minutes is given by \(\int _ { 0 } ^ { T } v \mathrm {~d} t\) Calculate the exact value of the displacement of the spacecraft over the first half minute given by the model.
  7. On further investigation the scientists believe the original model may be valid for up to three minutes. Explain why the approximation in (iii) will be no longer be valid for this time interval.
OCR MEI C4 Q9
17 marks Standard +0.3
9 Two astronomers wish to model the path of motion of a particle in two dimensions.
Experimental results show that the position of the particle can be found using the parametric equations $$x = 2 \cos \theta - \sin \theta + 2 \quad y = \cos \theta + 2 \sin \theta - 1 \quad \left( 0 \leq \theta \leq 360 ^ { \circ } \right)$$ One astronomer uses trigonometry.
  1. Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants to be determined. Show also that, for the same values of \(R\) and \(\alpha\), $$\cos \theta + 2 \sin \theta = R \sin ( \theta + \alpha )$$
  2. Hence, or otherwise, show that the path of particle may be written in the form $$( x - 2 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 5$$ Describe the path of the particle. The second astronomer sets up a first order differential equation with the condition that \(x = 4\) when \(y = 0\).
  3. Verify that the point with parameter \(\theta = 0\) has coordinates \(( 4,0 )\).
  4. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Deduce that \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x - 2 } { y + 1 }$$
  5. Solve this differential equation, using the condition that \(y = 0\) when \(x = 4\). Hence show that the two solutions give the same cartesian equation for the path of particle.
OCR C4 Q1
4 marks Moderate -0.3
  1. Express
$$\frac { 2 x } { 2 x ^ { 2 } + 3 x - 5 } \div \frac { x ^ { 3 } } { x ^ { 2 } - x }$$ as a single fraction in its simplest form.
OCR C4 Q2
7 marks Standard +0.3
2. A curve has the equation $$2 x ^ { 2 } + x y - y ^ { 2 } + 18 = 0$$ Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis.
OCR C4 Q3
8 marks Standard +0.3
3. The first four terms in the series expansion of \(( 1 + a x ) ^ { n }\) in ascending powers of \(x\) are $$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$ where \(a , n\) and \(k\) are constants and \(| a x | < 1\).
  1. Find the values of \(a\) and \(n\).
  2. Show that \(k = - 160\).
OCR C4 Q4
9 marks Standard +0.3
4. Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { c } 6 \\ 3 \\ - 6 \end{array} \right)\) respectively. Find, in exact, simplified form,
  1. the cosine of \(\angle A O B\),
  2. the area of triangle \(O A B\),
  3. the shortest distance from \(A\) to the line \(O B\).
OCR C4 Q5
10 marks Standard +0.3
5. (i) Use the derivatives of \(\sin x\) and \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$ The tangent to the curve \(y = 2 x \tan x\) at the point where \(x = \frac { \pi } { 4 }\) meets the \(y\)-axis at the point \(P\).
(ii) Find the \(y\)-coordinate of \(P\) in the form \(k \pi ^ { 2 }\) where \(k\) is a rational constant.
OCR C4 Q6
10 marks Standard +0.8
6. (i) Find $$\int \cot ^ { 2 } 2 x \mathrm {~d} x$$ (ii) Use the substitution \(u ^ { 2 } = x + 1\) to evaluate $$\int _ { 0 } ^ { 3 } \frac { x ^ { 2 } } { \sqrt { x + 1 } } \mathrm {~d} x$$
OCR C4 Q7
12 marks Standard +0.8
  1. During a chemical reaction, a compound is being made from two other substances.
At time \(t\) hours after the start of the reaction, \(x \mathrm {~g}\) of the compound has been produced. Assuming that \(x = 0\) initially, and that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = 2 ( x - 6 ) ( x - 3 )$$
  1. show that it takes approximately 7 minutes to produce 2 g of the compound.
  2. Explain why it is not possible to produce 3 g of the compound.
OCR C4 Q8
12 marks Standard +0.3
8. \includegraphics[max width=\textwidth, alt={}, center]{85427816-dcf1-49af-8d68-f4e88fc7d8f1-3_497_784_246_461} The diagram shows the curve with parametric equations $$x = - 1 + 4 \cos \theta , \quad y = 2 \sqrt { 2 } \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The point \(P\) on the curve has coordinates \(( 1 , \sqrt { 6 } )\).
  1. Find the value of \(\theta\) at \(P\).
  2. Show that the normal to the curve at \(P\) passes through the origin.
  3. Find a cartesian equation for the curve.
OCR C4 Q1
5 marks Standard +0.3
  1. A curve has the equation
$$x ^ { 2 } ( 2 + y ) - y ^ { 2 } = 0$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
OCR C4 Q2
5 marks Moderate -0.3
2. Show that $$\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x = 2 \ln 2 - \frac { 3 } { 4 }$$
OCR C4 Q3
7 marks Standard +0.8
3.
\includegraphics[max width=\textwidth, alt={}]{47c69f14-a336-4255-87fc-64ff1d2ee5e1-1_556_858_904_557}
The diagram shows the curve with equation \(y = 2 \sin x + \operatorname { cosec } x , 0 < x < \pi\).
The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 2 }\) is rotated through four right angles about the \(x\)-axis. Show that the volume of the solid formed is \(\frac { 1 } { 2 } \pi ( 4 \pi + 3 \sqrt { 3 } )\).
OCR C4 Q4
8 marks Moderate -0.8
4. (i) Express $$\frac { 4 x } { x ^ { 2 } - 9 } - \frac { 2 } { x + 3 }$$ as a single fraction in its simplest form.
(ii) Simplify $$\frac { x ^ { 3 } - 8 } { 3 x ^ { 2 } - 8 x + 4 }$$
OCR C4 Q5
11 marks Moderate -0.3
  1. A bath is filled with hot water which is allowed to cool. The temperature of the water is \(\theta ^ { \circ } \mathrm { C }\) after cooling for \(t\) minutes and the temperature of the room is assumed to remain constant at \(20 ^ { \circ } \mathrm { C }\).
Given that the rate at which the temperature of the water decreases is proportional to the difference in temperature between the water and the room,
  1. write down a differential equation connecting \(\theta\) and \(t\). Given also that the temperature of the water is initially \(37 ^ { \circ } \mathrm { C }\) and that it is \(36 ^ { \circ } \mathrm { C }\) after cooling for four minutes,
  2. find, to 3 significant figures, the temperature of the water after ten minutes. Advice suggests that the temperature of the water should be allowed to cool to \(33 ^ { \circ } \mathrm { C }\) before a child gets in.
  3. Find, to the nearest second, how long a child should wait before getting into the bath.
OCR C4 Q6
12 marks Standard +0.3
6. A curve has parametric equations $$x = 3 \cos ^ { 2 } t , \quad y = \sin 2 t , \quad 0 \leq t < \pi$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 } { 3 } \cot 2 t\).
  2. Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis.
  3. Show that the tangent to the curve at the point where \(t = \frac { \pi } { 6 }\) has the equation $$2 x + 3 \sqrt { 3 } y = 9$$
  4. Find a cartesian equation for the curve in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
OCR C4 Q7
12 marks Standard +0.8
7. Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } - 4 \\ 1 \\ 3 \end{array} \right)\) and \(\left( \begin{array} { c } - 3 \\ 6 \\ 1 \end{array} \right)\) respectively.
  1. Find a vector equation for the line \(l _ { 1 }\) which passes through \(A\) and \(B\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { c } 3 \\ - 7 \\ 9 \end{array} \right) + t \left( \begin{array} { c } 2 \\ - 3 \\ 1 \end{array} \right)$$
  2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
  3. Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).
OCR C4 Q9
Standard +0.3
9 \end{array} \right) + t \left( \begin{array} { c } 2
- 3
1 \end{array} \right)$$ (ii) Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
(iii) Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).
8. \(f ( x ) = \frac { 5 - 8 x } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }\).
(i) Express \(\mathrm { f } ( x )\) in partial fractions.
(ii) Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(iii) State the set of values of \(x\) for which your expansion is valid.
OCR C4 Q1
4 marks Moderate -0.8
  1. Evaluate
$$\int _ { 0 } ^ { \pi } \sin x ( 1 + \cos x ) d x$$
OCR C4 Q2
5 marks Moderate -0.8
  1. (i) Simplify
$$\frac { x ^ { 2 } + 7 x + 12 } { 2 x ^ { 2 } + 9 x + 4 }$$ (ii) Express $$\frac { x + 4 } { 2 x ^ { 2 } + 3 x + 1 } - \frac { 2 } { 2 x + 1 }$$ as a single fraction in its simplest form.
OCR C4 Q3
5 marks Standard +0.3
3. Find the exact value of $$\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x d x$$
OCR C4 Q4
6 marks Standard +0.3
4.
\includegraphics[max width=\textwidth, alt={}]{23bd8979-9ba6-4e77-a3d1-88feb5e5a5b3-1_444_728_1425_536}
The diagram shows the curve with parametric equations $$x = t + \sin t , \quad y = \sin t , \quad 0 \leq t \leq \pi$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find, in exact form, the coordinates of the point where the tangent to the curve is parallel to the \(x\)-axis.
OCR C4 Q5
7 marks Moderate -0.3
5. Given that \(y = - 2\) when \(x = 1\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ^ { 2 } \sqrt { x }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
OCR C4 Q6
9 marks Standard +0.3
6. (i) Find \(\int \tan ^ { 2 } 3 x \mathrm {~d} x\).
(ii) Using the substitution \(u = x ^ { 2 } + 4\), evaluate $$\int _ { 0 } ^ { 2 } \frac { 5 x } { \left( x ^ { 2 } + 4 \right) ^ { 2 } } d x$$
OCR C4 Q7
10 marks Standard +0.3
  1. A curve has the equation
$$3 x ^ { 2 } - 2 x + x y + y ^ { 2 } - 11 = 0$$ The point \(P\) on the curve has coordinates \(( - 1,3 )\).
  1. Show that the normal to the curve at \(P\) has the equation \(y = 2 - x\).
  2. Find the coordinates of the point where the normal to the curve at \(P\) meets the curve again.